Introduction In geometry, the problem of inscribing the largest possible regular polygon within a given shape is a classic problem that has been studied for centuries. In this article, we will explore the concept of inscribing the largest possible regular n-gon within a unit equilateral triangle, and then within the largest possible regular (n-1)-gon, and so on. We will derive a formula for the sum of the areas of these largest inscribed n-gons.
The Problem Consider a unit equilateral triangle with side length 1 unit. Inscribe the largest possible square within this triangle. Then, inscribe the largest possible regular pentagon within this square, and so on. We want to find the sum of the areas of these largest inscribed n-gons.
The Solution To solve this problem, we need to find the side length of the largest possible regular n-gon inscribed within the largest possible regular (n-1)-gon. Let's start with the case of n=3, where we have an equilateral triangle.
Case n=3 The largest possible square inscribed within an equilateral triangle has a side length of 3 2 \frac{\sqrt{3}}{2} 2 3 ( 3 2 ) 2 = 3 4 \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} ( 2 3 ) 2 = 4 3
Case n=4 The largest possible regular pentagon inscribed within a square has a side length of 1 2 \frac{1}{\sqrt{2}} 2 1 1 2 ⋅ 1 2 ⋅ 1 2 ⋅ 1 2 ⋅ 1 2 = 1 4 2 \frac{1}{2} \cdot \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = \frac{1}{4\sqrt{2}} 2 1 ⋅ 2 1 ⋅ 2 1 ⋅ 2 1 ⋅ 2 1 = 4 2 1
General Case Let's assume that the largest possible regular (n-1)-gon inscribed within a regular n-gon has a side length of s n − 1 s_{n-1} s n − 1 s n = s n − 1 sin ( π n ) s_n = \frac{s_{n-1}}{\sin\left(\frac{\pi}{n}\right)} s n = s i n ( n π ) s n − 1
The area of the largest possible regular n-gon inscribed within a regular (n-1)-gon is given by:
A n = s n 2 4 ⋅ sin ( 2 π n ) A_n = \frac{s_n^2}{4} \cdot \sin\left(\frac{2\pi}{n}\right)
A n = 4 s n 2 ⋅ sin ( n 2 π )
Substituting the expression for s n s_n s n
A n = ( s n − 1 sin ( π n ) ) 2 4 ⋅ sin ( 2 π n ) A_n = \frac{\left(\frac{s_{n-1}}{\sin\left(\frac{\pi}{n}\right)}\right)^2}{4} \cdot \sin\left(\frac{2\pi}{n}\right)
A n = 4 ( s i n ( n π ) s n − 1 ) 2 ⋅ sin ( n 2 π )
Simplifying this expression, we get:
A n = s n − 1 2 4 ⋅ sin ( 2 π n ) sin 2 ( π n ) A_n = \frac{s_{n-1}^2}{4} \cdot \frac{\sin\left(\frac{2\pi}{n}\right)}{\sin^2\left(\frac{\pi}{n}\right)}
A n = 4 s n − 1 2 ⋅ sin 2 ( n π ) sin ( n 2 π )
Now, let's find the sum of the areas of these largest inscribed n-gons.
Sum of Areas The sum of the areas of the largest inscribed n-gons is given by:
S = ∑ n = 3 ∞ A n S = \sum_{n=3}^{\infty} A_n
S = n = 3 ∑ ∞ A n
Substituting the expression for A n A_n A n
S = ∑ n = 3 ∞ s n − 1 2 4 ⋅ sin ( 2 π n ) sin 2 ( π n ) S = \sum_{n=3}^{\infty} \frac{s_{n-1}^2}{4} \cdot \frac{\sin\left(\frac{2\pi}{n}\right)}{\sin^2\left(\frac{\pi}{n}\right)}
S = n = 3 ∑ ∞ 4 s n − 1 2 ⋅ sin 2 ( n π ) sin ( n 2 π )
This is a series of fractions, where each fraction has a numerator that depends on the previous term in the series, and a denominator that depends on the current term in the series.
To evaluate this series, we can use the following trick:
∑ n = 3 ∞ s n − 1 2 4 ⋅ sin ( 2 π n ) sin 2 ( π n ) = ∑ n = 3 ∞ s n − 1 2 4 ⋅ sin ( 2 π n ) sin 2 ( π n ) ⋅ sin 2 ( π n ) sin 2 ( π n ) \sum_{n=3}^{\infty} \frac{s_{n-1}^2}{4} \cdot \frac{\sin\left(\frac{2\pi}{n}\right)}{\sin^2\left(\frac{\pi}{n}\right)} = \sum_{n=3}^{\infty} \frac{s_{n-1}^2}{4} \cdot \frac{\sin\left(\frac{2\pi}{n}\right)}{\sin^2\left(\frac{\pi}{n}\right)} \cdot \frac{\sin^2\left(\frac{\pi}{n}\right)}{\sin^2\left(\frac{\pi}{n}\right)}
n = 3 ∑ ∞ 4 s n − 1 2 ⋅ sin 2 ( n π ) sin ( n 2 π ) = n = 3 ∑ ∞ 4 s n − 1 2 ⋅ sin 2 ( n π ) sin ( n 2 π ) ⋅ sin 2 ( n π ) sin 2 ( n π )
Simplifying this expression, we get:
S = ∑ n = 3 ∞ s n − 1 2 4 ⋅ sin ( 2 π n ) sin 2 ( π n ) ⋅ sin 2 ( π n ) sin 2 ( π n ) S = \sum_{n=3}^{\infty} \frac{s_{n-1}^2}{4} \cdot \frac{\sin\left(\frac{2\pi}{n}\right)}{\sin^2\left(\frac{\pi}{n}\right)} \cdot \frac{\sin^2\left(\frac{\pi}{n}\right)}{\sin^2\left(\frac{\pi}{n}\right)}
S = n = 3 ∑ ∞ 4 s n − 1 2 ⋅ sin 2 ( n π ) sin ( n 2 π ) ⋅ sin 2 ( n π ) sin 2 ( n π )
Now, let's simplify the expression further.
Simplifying the Expression We can simplify the expression by using the following identity:
sin ( 2 π n ) = 2 sin ( π n ) cos ( π n ) \sin\left(\frac{2\pi}{n}\right) = 2\sin\left(\frac{\pi}{n}\right)\cos\left(\frac{\pi}{n}\right)
sin ( n 2 π ) = 2 sin ( n π ) cos ( n π )
Substituting this identity, we get:
S = ∑ n = 3 ∞ s n − 1 2 4 ⋅ 2 sin ( π n ) cos ( π n ) sin 2 ( π n ) ⋅ sin 2 ( π n ) sin 2 ( π n ) S = \sum_{n=3}^{\infty} \frac{s_{n-1}^2}{4} \cdot \frac{2\sin\left(\frac{\pi}{n}\right)\cos\left(\frac{\pi}{n}\right)}{\sin^2\left(\frac{\pi}{n}\right)} \cdot \frac{\sin^2\left(\frac{\pi}{n}\right)}{\sin^2\left(\frac{\pi}{n}\right)}
S = n = 3 ∑ ∞ 4 s n − 1 2 ⋅ sin 2 ( n π ) 2 sin ( n π ) cos ( n π ) ⋅ sin 2 ( n π ) sin 2 ( n π )
Simplifying this expression, we get:
S = ∑ n = 3 ∞ s n − 1 2 2 ⋅ cos ( π n ) sin ( π n ) S = \sum_{n=3}^{\infty} \frac{s_{n-1}^2}{2} \cdot \frac{\cos\left(\frac{\pi}{n}\right)}{\sin\left(\frac{\pi}{n}\right)}
S = n = 3 ∑ ∞ 2 s n − 1 2 ⋅ sin ( n π ) cos ( n π )
Now, let's simplify the expression further.
Simplifying the Expression Further We can simplify the expression by using the following identity:
cos ( π n ) = sin ( π n ) ⋅ cot ( π n ) \cos\left(\frac{\pi}{n}\right) = \sin\left(\frac{\pi}{n}\right) \cdot \cot\left(\frac{\pi}{n}\right)
cos ( n π ) = sin ( n π ) ⋅ cot ( n π )
Substituting this identity, we get:
S = ∑ n = 3 ∞ s n − 1 2 2 ⋅ sin ( π n ) ⋅ cot ( π n ) sin ( π n ) S = \sum_{n=3}^{\infty} \frac{s_{n-1}^2}{2} \cdot \frac{\sin\left(\frac{\pi}{n}\right) \cdot \cot\left(\frac{\pi}{n}\right)}{\sin\left(\frac{\pi}{n}\right)}
S = n = 3 ∑ ∞ 2 s n − 1 2 ⋅ sin ( n π ) sin ( n π ) ⋅ cot ( n π )
Simplifying this expression, we get:
S = ∑ n = 3 ∞ s n − 1 2 2 ⋅ cot ( π n ) S = \sum_{n=3}^{\infty} \frac{s_{n-1}^2}{2} \cdot \cot\left(\frac{\pi}{n}\right)
S = n = 3 ∑ ∞ 2 s n − 1 2 ⋅ cot ( n π )
Now, let's simplify the expression further.
Simplifying the Expression Further We can simplify the expression by using the following identity:
cot ( π n ) = 1 tan ( π n ) \cot\left(\frac{\pi}{n}\right) = \frac{1}{\tan\left(\frac{\pi}{n}\right)}
cot ( n π ) = tan ( n π ) 1
Substituting this identity, we get:
S = ∑ n = 3 ∞ s n − 1 2 2 ⋅ 1 tan ( π n ) S = \sum_{n=3}^{\infty} \frac{s_{n-1}^2}{2} \cdot \frac{1}{\tan\left(\frac{\pi}{n}\right)}
S = n = 3 ∑ ∞ 2 s n − 1 2 ⋅ tan ( n π ) 1
Simplifying this expression, we get:
S = ∑ n = 3 ∞ s n − 1 2 2 sin ( π n ) cos ( π n ) S = \sum_{n=3}^{\infty} \frac{s_{n-1}^2}{2\sin\left(\frac{\pi}{n}\right)\cos\left(\frac{\pi}{n}\right)}
S = n = 3 ∑ ∞ 2 sin ( n π ) cos ( n π ) s n − 1 2
Now, let's simplify the expression further.
Simplifying the Expression Further We can simplify the expression by using the following identity:
sin ( π n ) cos ( π n ) = 1 2 sin ( 2 π n ) \sin\left(\frac{\pi}{n}\right)\cos\left(\frac{\pi}{n}\right) = \frac{1}{2}\sin\left(\frac{2\pi}{n}\right)
sin ( n π ) cos ( n π ) = 2 1 sin ( n 2 π )
Substituting this identity, we get:
S = ∑ n = 3 ∞ s n − 1 2 sin ( 2 π n ) S = \sum_{n=3}^{\infty} \frac{s_{n-1}^2}{\sin\left(\frac{2\pi}{n}\right)}
S = n = 3 ∑ ∞ sin ( n 2 π ) s n − 1 2
Now, let's simplify the expression further.
Simplifying the Expression Further We can simplify the expression by using the following identity:
sin ( 2 π n ) = 2 sin ( π n ) cos ( π n ) \sin\left(\frac{2\pi}{n}\right) = 2\sin\left(\frac{\pi}{n}\right)\cos\left(\frac{\pi}{n}\right)
sin ( n 2 π ) = 2 sin ( n π ) cos ( n π )
Substituting this identity, we get:
S = \sum_{n=3}^{\infty} \frac{s_{n-1}^2}{2\<br/>
**Q&A: The Sum of Largest Inscribed N-gons**
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Q: What is the problem of inscribing the largest possible regular n-gon within a given shape? A: The problem of inscribing the largest possible regular n-gon within a given shape is a classic problem in geometry that has been studied for centuries. It involves finding the largest possible regular n-gon that can be inscribed within a given shape, such as a unit equilateral triangle.
Q: How do you find the side length of the largest possible regular n-gon inscribed within the largest possible regular (n-1)-gon? A: To find the side length of the largest possible regular n-gon inscribed within the largest possible regular (n-1)-gon, we can use the following formula:
s n = s n − 1 sin ( π n ) < / s p a n > < / p > < p > w h e r e < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u b > < m i > s < / m i > < m i > n < / m i > < / m s u b > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > s n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.5806 e m ; v e r t i c a l − a l i g n : − 0.15 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > s < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s t h e s i d e l e n g t h o f t h e l a r g e s t p o s s i b l e r e g u l a r n − g o n , a n d < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u b > < m i > s < / m i > < m r o w > < m i > n < / m i > < m o > − < / m o > < m n > 1 < / m n > < / m r o w > < / m s u b > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > s n − 1 < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6389 e m ; v e r t i c a l − a l i g n : − 0.2083 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > s < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.3011 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < s p a n c l a s s = " m b i n m t i g h t " > − < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2083 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s t h e s i d e l e n g t h o f t h e l a r g e s t p o s s i b l e r e g u l a r ( n − 1 ) − g o n . < / p > < h 2 > < s t r o n g > Q : W h a t i s t h e a r e a o f t h e l a r g e s t p o s s i b l e r e g u l a r n − g o n i n s c r i b e d w i t h i n a r e g u l a r ( n − 1 ) − g o n ? < / s t r o n g > < / h 2 > < p > A : T h e a r e a o f t h e l a r g e s t p o s s i b l e r e g u l a r n − g o n i n s c r i b e d w i t h i n a r e g u l a r ( n − 1 ) − g o n i s g i v e n b y : < / p > < p c l a s s = ′ k a t e x − b l o c k ′ > < s p a n c l a s s = " k a t e x − d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m s u b > < m i > A < / m i > < m i > n < / m i > < / m s u b > < m o > = < / m o > < m f r a c > < m s u b s u p > < m i > s < / m i > < m i > n < / m i > < m n > 2 < / m n > < / m s u b s u p > < m n > 4 < / m n > < / m f r a c > < m o > ⋅ < / m o > < m i > s i n < / m i > < m o > < / m o > < m r o w > < m o f e n c e = " t r u e " > ( < / m o > < m f r a c > < m r o w > < m n > 2 < / m n > < m i > π < / m i > < / m r o w > < m i > n < / m i > < / m f r a c > < m o f e n c e = " t r u e " > ) < / m o > < / m r o w > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > A n = s n 2 4 ⋅ sin ( 2 π n ) < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.8333 e m ; v e r t i c a l − a l i g n : − 0.15 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > A < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.1771 e m ; v e r t i c a l − a l i g n : − 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.4911 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > s < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.8141 e m ; " > < s p a n s t y l e = " t o p : − 2.453 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.063 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 2 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > ⋅ < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.4 e m ; v e r t i c a l − a l i g n : − 0.95 e m ; " > < / s p a n > < s p a n c l a s s = " m o p " > s i n < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m i n n e r " > < s p a n c l a s s = " m o p e n d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ( < / s p a n > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > n < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.03588 e m ; " > π < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e d e l i m c e n t e r " s t y l e = " t o p : 0 e m ; " > < s p a n c l a s s = " d e l i m s i z i n g s i z e 3 " > ) < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < p > w h e r e < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u b > < m i > A < / m i > < m i > n < / m i > < / m s u b > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > A n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.8333 e m ; v e r t i c a l − a l i g n : − 0.15 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > A < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s t h e a r e a o f t h e l a r g e s t p o s s i b l e r e g u l a r n − g o n , a n d < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u b > < m i > s < / m i > < m i > n < / m i > < / m s u b > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > s n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.5806 e m ; v e r t i c a l − a l i g n : − 0.15 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > s < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s t h e s i d e l e n g t h o f t h e l a r g e s t p o s s i b l e r e g u l a r n − g o n . < / p > < h 2 > < s t r o n g > Q : H o w d o y o u f i n d t h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s ? < / s t r o n g > < / h 2 > < p > A : T o f i n d t h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s , w e c a n u s e t h e f o l l o w i n g f o r m u l a : < / p > < p c l a s s = ′ k a t e x − b l o c k ′ > < s p a n c l a s s = " k a t e x − d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m i > S < / m i > < m o > = < / m o > < m u n d e r o v e r > < m o > ∑ < / m o > < m r o w > < m i > n < / m i > < m o > = < / m o > < m n > 3 < / m n > < / m r o w > < m i m a t h v a r i a n t = " n o r m a l " > ∞ < / m i > < / m u n d e r o v e r > < m s u b > < m i > A < / m i > < m i > n < / m i > < / m s u b > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > S = ∑ n = 3 ∞ A n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.05764 e m ; " > S < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.9185 e m ; v e r t i c a l − a l i g n : − 1.2671 e m ; " > < / s p a n > < s p a n c l a s s = " m o p o p − l i m i t s " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.6514 e m ; " > < s p a n s t y l e = " t o p : − 1.8829 e m ; m a r g i n − l e f t : 0 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3.05 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < s p a n c l a s s = " m r e l m t i g h t " > = < / s p a n > < s p a n c l a s s = " m o r d m t i g h t " > 3 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3.05 e m ; " > < / s p a n > < s p a n > < s p a n c l a s s = " m o p o p − s y m b o l l a r g e − o p " > ∑ < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 4.3 e m ; m a r g i n − l e f t : 0 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3.05 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > ∞ < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.2671 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > A < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < p > w h e r e < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m i > S < / m i > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > S < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.05764 e m ; " > S < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s t h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s , a n d < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " > < s e m a n t i c s > < m r o w > < m s u b > < m i > A < / m i > < m i > n < / m i > < / m s u b > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > A n < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.8333 e m ; v e r t i c a l − a l i g n : − 0.15 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d m a t h n o r m a l " > A < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " > < s p a n s t y l e = " t o p : − 2.55 e m ; m a r g i n − l e f t : 0 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m a t h n o r m a l m t i g h t " > n < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > i s t h e a r e a o f t h e l a r g e s t p o s s i b l e r e g u l a r n − g o n . < / p > < h 2 > < s t r o n g > Q : W h a t i s t h e v a l u e o f t h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s ? < / s t r o n g > < / h 2 > < p > A : T h e v a l u e o f t h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s i s g i v e n b y : < / p > < p c l a s s = ′ k a t e x − b l o c k ′ > < s p a n c l a s s = " k a t e x − d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m i > S < / m i > < m o > = < / m o > < m f r a c > < m n > 3 < / m n > < m n > 4 < / m n > < / m f r a c > < m o > + < / m o > < m f r a c > < m n > 1 < / m n > < m r o w > < m n > 4 < / m n > < m s q r t > < m n > 2 < / m n > < / m s q r t > < / m r o w > < / m f r a c > < m o > + < / m o > < m f r a c > < m n > 1 < / m n > < m r o w > < m n > 4 < / m n > < m s q r t > < m n > 3 < / m n > < / m s q r t > < / m r o w > < / m f r a c > < m o > + < / m o > < m f r a c > < m n > 1 < / m n > < m r o w > < m n > 4 < / m n > < m s q r t > < m n > 4 < / m n > < / m s q r t > < / m r o w > < / m f r a c > < m o > + < / m o > < m f r a c > < m n > 1 < / m n > < m r o w > < m n > 4 < / m n > < m s q r t > < m n > 5 < / m n > < / m s q r t > < / m r o w > < / m f r a c > < m o > + < / m o > < m f r a c > < m n > 1 < / m n > < m r o w > < m n > 4 < / m n > < m s q r t > < m n > 6 < / m n > < / m s q r t > < / m r o w > < / m f r a c > < m o > + < / m o > < m o > ⋯ < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > S = 3 4 + 1 4 2 + 1 4 3 + 1 4 4 + 1 4 5 + 1 4 6 + ⋯ < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " s t y l e = " m a r g i n − r i g h t : 0.05764 e m ; " > S < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v e r t i c a l − a l i g n : − 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 3 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.2514 e m ; v e r t i c a l − a l i g n : − 0.93 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.2028 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < s p a n c l a s s = " m o r d s q r t " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9072 e m ; " > < s p a n c l a s s = " s v g − a l i g n " s t y l e = " t o p : − 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " s t y l e = " p a d d i n g − l e f t : 0.833 e m ; " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 2.8672 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e − t a i l " s t y l e = " m i n − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.08 e m " v i e w B o x = " 004000001080 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1328 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.93 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.2514 e m ; v e r t i c a l − a l i g n : − 0.93 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.2028 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < s p a n c l a s s = " m o r d s q r t " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9072 e m ; " > < s p a n c l a s s = " s v g − a l i g n " s t y l e = " t o p : − 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " s t y l e = " p a d d i n g − l e f t : 0.833 e m ; " > < s p a n c l a s s = " m o r d " > 3 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 2.8672 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e − t a i l " s t y l e = " m i n − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.08 e m " v i e w B o x = " 004000001080 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1328 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.93 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.2514 e m ; v e r t i c a l − a l i g n : − 0.93 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.2028 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < s p a n c l a s s = " m o r d s q r t " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9072 e m ; " > < s p a n c l a s s = " s v g − a l i g n " s t y l e = " t o p : − 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " s t y l e = " p a d d i n g − l e f t : 0.833 e m ; " > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 2.8672 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e − t a i l " s t y l e = " m i n − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.08 e m " v i e w B o x = " 004000001080 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1328 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.93 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.2514 e m ; v e r t i c a l − a l i g n : − 0.93 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.2028 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < s p a n c l a s s = " m o r d s q r t " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9072 e m ; " > < s p a n c l a s s = " s v g − a l i g n " s t y l e = " t o p : − 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " s t y l e = " p a d d i n g − l e f t : 0.833 e m ; " > < s p a n c l a s s = " m o r d " > 5 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 2.8672 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e − t a i l " s t y l e = " m i n − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.08 e m " v i e w B o x = " 004000001080 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1328 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.93 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.2514 e m ; v e r t i c a l − a l i g n : − 0.93 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.2028 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 4 < / s p a n > < s p a n c l a s s = " m o r d s q r t " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.9072 e m ; " > < s p a n c l a s s = " s v g − a l i g n " s t y l e = " t o p : − 3 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " s t y l e = " p a d d i n g − l e f t : 0.833 e m ; " > < s p a n c l a s s = " m o r d " > 6 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 2.8672 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " h i d e − t a i l " s t y l e = " m i n − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " > < s v g x m l n s = " h t t p : / / w w w . w 3. o r g / 2000 / s v g " w i d t h = " 400 e m " h e i g h t = " 1.08 e m " v i e w B o x = " 004000001080 " p r e s e r v e A s p e c t R a t i o = " x M i n Y M i n s l i c e " > < p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z " / > < / s v g > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.1328 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.93 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 0.313 e m ; " > < / s p a n > < s p a n c l a s s = " m i n n e r " > ⋯ < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < p > T h i s i s a n i n f i n i t e s e r i e s , a n d i t s v a l u e i s n o t k n o w n e x a c t l y . < / p > < h 2 > < s t r o n g > Q : W h a t i s t h e s i g n i f i c a n c e o f t h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s ? < / s t r o n g > < / h 2 > < p > A : T h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s i s a f u n d a m e n t a l p r o b l e m i n g e o m e t r y t h a t h a s b e e n s t u d i e d f o r c e n t u r i e s . I t h a s m a n y a p p l i c a t i o n s i n m a t h e m a t i c s , p h y s i c s , a n d e n g i n e e r i n g , a n d i s a k e y c o n c e p t i n t h e s t u d y o f g e o m e t r i c s h a p e s a n d t h e i r p r o p e r t i e s . < / p > < h 2 > < s t r o n g > Q : H o w c a n I u s e t h e s u m o f t h e a r e a s o f l a r g e s t i n s c r i b e d n − g o n s i n r e a l − w o r l d a p p l i c a t i o n s ? < / s t r o n g > < / h 2 > < p > A : T h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s c a n b e u s e d i n m a n y r e a l − w o r l d a p p l i c a t i o n s , s u c h a s : < / p > < u l > < l i > < s t r o n g > C o m p u t e r − a i d e d d e s i g n ( C A D ) < / s t r o n g > : T h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s c a n b e u s e d t o c r e a t e c o m p l e x g e o m e t r i c s h a p e s a n d m o d e l s . < / l i > < l i > < s t r o n g > E n g i n e e r i n g < / s t r o n g > : T h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s c a n b e u s e d t o d e s i g n a n d o p t i m i z e g e o m e t r i c s h a p e s a n d s t r u c t u r e s . < / l i > < l i > < s t r o n g > P h y s i c s < / s t r o n g > : T h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s c a n b e u s e d t o s t u d y t h e p r o p e r t i e s o f g e o m e t r i c s h a p e s a n d t h e i r b e h a v i o r u n d e r d i f f e r e n t c o n d i t i o n s . < / l i > < / u l > < h 2 > < s t r o n g > Q : W h a t a r e s o m e c o m m o n m i s t a k e s t o a v o i d w h e n w o r k i n g w i t h t h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s ? < / s t r o n g > < / h 2 > < p > A : S o m e c o m m o n m i s t a k e s t o a v o i d w h e n w o r k i n g w i t h t h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s i n c l u d e : < / p > < u l > < l i > < s t r o n g > I n c o r r e c t l y c a l c u l a t i n g t h e s i d e l e n g t h o f t h e l a r g e s t p o s s i b l e r e g u l a r n − g o n < / s t r o n g > : M a k e s u r e t o u s e t h e c o r r e c t f o r m u l a t o c a l c u l a t e t h e s i d e l e n g t h o f t h e l a r g e s t p o s s i b l e r e g u l a r n − g o n . < / l i > < l i > < s t r o n g > I n c o r r e c t l y c a l c u l a t i n g t h e a r e a o f t h e l a r g e s t p o s s i b l e r e g u l a r n − g o n < / s t r o n g > : M a k e s u r e t o u s e t h e c o r r e c t f o r m u l a t o c a l c u l a t e t h e a r e a o f t h e l a r g e s t p o s s i b l e r e g u l a r n − g o n . < / l i > < l i > < s t r o n g > N o t c o n s i d e r i n g t h e i n f i n i t e s e r i e s < / s t r o n g > : T h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s i s a n i n f i n i t e s e r i e s , a n d i t s v a l u e i s n o t k n o w n e x a c t l y . M a k e s u r e t o c o n s i d e r t h e i n f i n i t e s e r i e s w h e n w o r k i n g w i t h t h e s u m o f t h e a r e a s o f t h e l a r g e s t i n s c r i b e d n − g o n s . < / l i > < / u l > s_n = \frac{s_{n-1}}{\sin\left(\frac{\pi}{n}\right)}
</span></p>
<p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">s_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the side length of the largest possible regular n-gon, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">s_{n-1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6389em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span></span></span></span> is the side length of the largest possible regular (n-1)-gon.</p>
<h2><strong>Q: What is the area of the largest possible regular n-gon inscribed within a regular (n-1)-gon?</strong></h2>
<p>A: The area of the largest possible regular n-gon inscribed within a regular (n-1)-gon is given by:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>A</mi><mi>n</mi></msub><mo>=</mo><mfrac><msubsup><mi>s</mi><mi>n</mi><mn>2</mn></msubsup><mn>4</mn></mfrac><mo>⋅</mo><mi>sin</mi><mo></mo><mrow><mo fence="true">(</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>n</mi></mfrac><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">A_n = \frac{s_n^2}{4} \cdot \sin\left(\frac{2\pi}{n}\right)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1771em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4911em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="mop">sin</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">n</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span><span class="mord mathnormal" style="margin-right:0.03588em;">π</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p>
<p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the area of the largest possible regular n-gon, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>s</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">s_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5806em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">s</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the side length of the largest possible regular n-gon.</p>
<h2><strong>Q: How do you find the sum of the areas of the largest inscribed n-gons?</strong></h2>
<p>A: To find the sum of the areas of the largest inscribed n-gons, we can use the following formula:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>S</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><mi mathvariant="normal">∞</mi></munderover><msub><mi>A</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S = \sum_{n=3}^{\infty} A_n
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.9185em;vertical-align:-1.2671em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6514em;"><span style="top:-1.8829em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mrel mtight">=</span><span class="mord mtight">3</span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∞</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2671em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span></span></span></span> is the sum of the areas of the largest inscribed n-gons, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>A</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> is the area of the largest possible regular n-gon.</p>
<h2><strong>Q: What is the value of the sum of the areas of the largest inscribed n-gons?</strong></h2>
<p>A: The value of the sum of the areas of the largest inscribed n-gons is given by:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>S</mi><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><mn>3</mn></msqrt></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><mn>4</mn></msqrt></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><mn>5</mn></msqrt></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><mn>6</mn></msqrt></mrow></mfrac><mo>+</mo><mo>⋯</mo></mrow><annotation encoding="application/x-tex">S = \frac{3}{4} + \frac{1}{4\sqrt{2}} + \frac{1}{4\sqrt{3}} + \frac{1}{4\sqrt{4}} + \frac{1}{4\sqrt{5}} + \frac{1}{4\sqrt{6}} + \cdots
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.2514em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.2028em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">4</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9072em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.8672em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702
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<p>This is an infinite series, and its value is not known exactly.</p>
<h2><strong>Q: What is the significance of the sum of the areas of the largest inscribed n-gons?</strong></h2>
<p>A: The sum of the areas of the largest inscribed n-gons is a fundamental problem in geometry that has been studied for centuries. It has many applications in mathematics, physics, and engineering, and is a key concept in the study of geometric shapes and their properties.</p>
<h2><strong>Q: How can I use the sum of the areas of largest inscribed n-gons in real-world applications?</strong></h2>
<p>A: The sum of the areas of the largest inscribed n-gons can be used in many real-world applications, such as:</p>
<ul>
<li><strong>Computer-aided design (CAD)</strong>: The sum of the areas of the largest inscribed n-gons can be used to create complex geometric shapes and models.</li>
<li><strong>Engineering</strong>: The sum of the areas of the largest inscribed n-gons can be used to design and optimize geometric shapes and structures.</li>
<li><strong>Physics</strong>: The sum of the areas of the largest inscribed n-gons can be used to study the properties of geometric shapes and their behavior under different conditions.</li>
</ul>
<h2><strong>Q: What are some common mistakes to avoid when working with the sum of the areas of the largest inscribed n-gons?</strong></h2>
<p>A: Some common mistakes to avoid when working with the sum of the areas of the largest inscribed n-gons include:</p>
<ul>
<li><strong>Incorrectly calculating the side length of the largest possible regular n-gon</strong>: Make sure to use the correct formula to calculate the side length of the largest possible regular n-gon.</li>
<li><strong>Incorrectly calculating the area of the largest possible regular n-gon</strong>: Make sure to use the correct formula to calculate the area of the largest possible regular n-gon.</li>
<li><strong>Not considering the infinite series</strong>: The sum of the areas of the largest inscribed n-gons is an infinite series, and its value is not known exactly. Make sure to consider the infinite series when working with the sum of the areas of the largest inscribed n-gons.</li>
</ul>
s n = sin ( n π ) s n − 1 < / s p an >< / p >< p > w h ere < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u b >< mi > s < / mi >< mi > n < / mi >< / m s u b >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > s n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.5806 e m ; v er t i c a l − a l i g n : − 0.15 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > s < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > i s t h es i d e l e n g t h o f t h e l a r g es tp oss ib l ere gu l a r n − g o n , an d < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u b >< mi > s < / mi >< m ro w >< mi > n < / mi >< m o > − < / m o >< mn > 1 < / mn >< / m ro w >< / m s u b >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > s n − 1 < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6389 e m ; v er t i c a l − a l i g n : − 0.2083 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > s < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.3011 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< s p an c l a ss = " mbinm t i g h t " > − < / s p an >< s p an c l a ss = " m or d m t i g h t " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2083 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > i s t h es i d e l e n g t h o f t h e l a r g es tp oss ib l ere gu l a r ( n − 1 ) − g o n . < / p >< h 2 >< s t ro n g > Q : Wha t i s t h e a re a o f t h e l a r g es tp oss ib l ere gu l a r n − g o nin scr ib e d w i t hina re gu l a r ( n − 1 ) − g o n ? < / s t ro n g >< / h 2 >< p > A : T h e a re a o f t h e l a r g es tp oss ib l ere gu l a r n − g o nin scr ib e d w i t hina re gu l a r ( n − 1 ) − g o ni s g i v e nb y :< / p >< p c l a ss = ′ ka t e x − b l oc k ′ >< s p an c l a ss = " ka t e x − d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m s u b >< mi > A < / mi >< mi > n < / mi >< / m s u b >< m o >=< / m o >< m f r a c >< m s u b s u p >< mi > s < / mi >< mi > n < / mi >< mn > 2 < / mn >< / m s u b s u p >< mn > 4 < / mn >< / m f r a c >< m o > ⋅ < / m o >< mi > s in < / mi >< m o > < / m o >< m ro w >< m o f e n ce = " t r u e " > ( < / m o >< m f r a c >< m ro w >< mn > 2 < / mn >< mi > π < / mi >< / m ro w >< mi > n < / mi >< / m f r a c >< m o f e n ce = " t r u e " > ) < / m o >< / m ro w >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > A n = 4 s n 2 ⋅ sin ( n 2 π ) < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.8333 e m ; v er t i c a l − a l i g n : − 0.15 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > A < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.1771 e m ; v er t i c a l − a l i g n : − 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.4911 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 4 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > s < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.8141 e m ; " >< s p an s t y l e = " t o p : − 2.453 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.063 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 2 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.247 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > ⋅ < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.4 e m ; v er t i c a l − a l i g n : − 0.95 e m ; " >< / s p an >< s p an c l a ss = " m o p " > s in < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " minn er " >< s p an c l a ss = " m o p e n d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ( < / s p an >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > n < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.03588 e m ; " > π < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m c l ose d e l im ce n t er " s t y l e = " t o p : 0 e m ; " >< s p an c l a ss = " d e l im s i z in g s i ze 3" > ) < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< p > w h ere < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u b >< mi > A < / mi >< mi > n < / mi >< / m s u b >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > A n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.8333 e m ; v er t i c a l − a l i g n : − 0.15 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > A < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > i s t h e a re a o f t h e l a r g es tp oss ib l ere gu l a r n − g o n , an d < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u b >< mi > s < / mi >< mi > n < / mi >< / m s u b >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > s n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.5806 e m ; v er t i c a l − a l i g n : − 0.15 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > s < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > i s t h es i d e l e n g t h o f t h e l a r g es tp oss ib l ere gu l a r n − g o n . < / p >< h 2 >< s t ro n g > Q : Ho w d oyo u f in d t h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n s ? < / s t ro n g >< / h 2 >< p > A : T o f in d t h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n s , w ec an u se t h e f o ll o w in g f or m u l a :< / p >< p c l a ss = ′ ka t e x − b l oc k ′ >< s p an c l a ss = " ka t e x − d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< mi > S < / mi >< m o >=< / m o >< m u n d ero v er >< m o > ∑ < / m o >< m ro w >< mi > n < / mi >< m o >=< / m o >< mn > 3 < / mn >< / m ro w >< mima t h v a r ian t = " n or ma l " > ∞ < / mi >< / m u n d ero v er >< m s u b >< mi > A < / mi >< mi > n < / mi >< / m s u b >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > S = n = 3 ∑ ∞ A n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.05764 e m ; " > S < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.9185 e m ; v er t i c a l − a l i g n : − 1.2671 e m ; " >< / s p an >< s p an c l a ss = " m o p o p − l imi t s " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.6514 e m ; " >< s p an s t y l e = " t o p : − 1.8829 e m ; ma r g in − l e f t : 0 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3.05 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< s p an c l a ss = " m re l m t i g h t " >=< / s p an >< s p an c l a ss = " m or d m t i g h t " > 3 < / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3.05 e m ; " >< / s p an >< s p an >< s p an c l a ss = " m o p o p − sy mb o ll a r g e − o p " > ∑ < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 4.3 e m ; ma r g in − l e f t : 0 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3.05 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > ∞ < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.2671 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > A < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< p > w h ere < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< mi > S < / mi >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > S < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.05764 e m ; " > S < / s p an >< / s p an >< / s p an >< / s p an > i s t h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n s , an d < s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " >< se man t i cs >< m ro w >< m s u b >< mi > A < / mi >< mi > n < / mi >< / m s u b >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > A n < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.8333 e m ; v er t i c a l − a l i g n : − 0.15 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d ma t hn or ma l " > A < / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1514 e m ; " >< s p an s t y l e = " t o p : − 2.55 e m ; ma r g in − l e f t : 0 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d ma t hn or ma l m t i g h t " > n < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.15 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an > i s t h e a re a o f t h e l a r g es tp oss ib l ere gu l a r n − g o n . < / p >< h 2 >< s t ro n g > Q : Wha t i s t h e v a l u eo f t h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n s ? < / s t ro n g >< / h 2 >< p > A : T h e v a l u eo f t h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n s i s g i v e nb y :< / p >< p c l a ss = ′ ka t e x − b l oc k ′ >< s p an c l a ss = " ka t e x − d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< mi > S < / mi >< m o >=< / m o >< m f r a c >< mn > 3 < / mn >< mn > 4 < / mn >< / m f r a c >< m o > + < / m o >< m f r a c >< mn > 1 < / mn >< m ro w >< mn > 4 < / mn >< m s q r t >< mn > 2 < / mn >< / m s q r t >< / m ro w >< / m f r a c >< m o > + < / m o >< m f r a c >< mn > 1 < / mn >< m ro w >< mn > 4 < / mn >< m s q r t >< mn > 3 < / mn >< / m s q r t >< / m ro w >< / m f r a c >< m o > + < / m o >< m f r a c >< mn > 1 < / mn >< m ro w >< mn > 4 < / mn >< m s q r t >< mn > 4 < / mn >< / m s q r t >< / m ro w >< / m f r a c >< m o > + < / m o >< m f r a c >< mn > 1 < / mn >< m ro w >< mn > 4 < / mn >< m s q r t >< mn > 5 < / mn >< / m s q r t >< / m ro w >< / m f r a c >< m o > + < / m o >< m f r a c >< mn > 1 < / mn >< m ro w >< mn > 4 < / mn >< m s q r t >< mn > 6 < / mn >< / m s q r t >< / m ro w >< / m f r a c >< m o > + < / m o >< m o > ⋯ < / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > S = 4 3 + 4 2 1 + 4 3 1 + 4 4 1 + 4 5 1 + 4 6 1 + ⋯ < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.6833 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " s t y l e = " ma r g in − r i g h t : 0.05764 e m ; " > S < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v er t i c a l − a l i g n : − 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 4 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 3 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.2514 e m ; v er t i c a l − a l i g n : − 0.93 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.2028 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 4 < / s p an >< s p an c l a ss = " m or d s q r t " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9072 e m ; " >< s p an c l a ss = " s vg − a l i g n " s t y l e = " t o p : − 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " s t y l e = " p a dd in g − l e f t : 0.833 e m ; " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 2.8672 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " hi d e − t ai l " s t y l e = " min − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.08 e m " v i e wB o x = "004000001080" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1328 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.93 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.2514 e m ; v er t i c a l − a l i g n : − 0.93 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.2028 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 4 < / s p an >< s p an c l a ss = " m or d s q r t " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9072 e m ; " >< s p an c l a ss = " s vg − a l i g n " s t y l e = " t o p : − 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " s t y l e = " p a dd in g − l e f t : 0.833 e m ; " >< s p an c l a ss = " m or d " > 3 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 2.8672 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " hi d e − t ai l " s t y l e = " min − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.08 e m " v i e wB o x = "004000001080" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1328 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.93 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.2514 e m ; v er t i c a l − a l i g n : − 0.93 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.2028 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 4 < / s p an >< s p an c l a ss = " m or d s q r t " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9072 e m ; " >< s p an c l a ss = " s vg − a l i g n " s t y l e = " t o p : − 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " s t y l e = " p a dd in g − l e f t : 0.833 e m ; " >< s p an c l a ss = " m or d " > 4 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 2.8672 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " hi d e − t ai l " s t y l e = " min − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.08 e m " v i e wB o x = "004000001080" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1328 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.93 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.2514 e m ; v er t i c a l − a l i g n : − 0.93 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.2028 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 4 < / s p an >< s p an c l a ss = " m or d s q r t " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9072 e m ; " >< s p an c l a ss = " s vg − a l i g n " s t y l e = " t o p : − 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " s t y l e = " p a dd in g − l e f t : 0.833 e m ; " >< s p an c l a ss = " m or d " > 5 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 2.8672 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " hi d e − t ai l " s t y l e = " min − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.08 e m " v i e wB o x = "004000001080" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1328 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.93 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.2514 e m ; v er t i c a l − a l i g n : − 0.93 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.2028 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 4 < / s p an >< s p an c l a ss = " m or d s q r t " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.9072 e m ; " >< s p an c l a ss = " s vg − a l i g n " s t y l e = " t o p : − 3 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " s t y l e = " p a dd in g − l e f t : 0.833 e m ; " >< s p an c l a ss = " m or d " > 6 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 2.8672 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " hi d e − t ai l " s t y l e = " min − w i d t h : 0.853 e m ; h e i g h t : 1.08 e m ; " >< s vgx m l n s = " h ttp : // www . w 3. or g /2000/ s vg " w i d t h = "400 e m " h e i g h t = "1.08 e m " v i e wB o x = "004000001080" p reser v e A s p ec tR a t i o = " x M inY M in s l i ce " >< p a t h d = " M 95 , 702 c − 2.7 , 0 , − 7.17 , − 2.7 , − 13.5 , − 8 c − 5.8 , − 5.3 , − 9.5 , − 10 , − 9.5 , − 14 c 0 , − 2 , 0.3 , − 3.3 , 1 , − 4 c 1.3 , − 2.7 , 23.83 , − 20.7 , 67.5 , − 54 c 44.2 , − 33.3 , 65.8 , − 50.3 , 66.5 , − 51 c 1.3 , − 1.3 , 3 , − 2 , 5 , − 2 c 4.7 , 0 , 8.7 , 3.3 , 12 , 10 s 173 , 378 , 173 , 378 c 0.7 , 0 , 35.3 , − 71 , 104 , − 213 c 68.7 , − 142 , 137.5 , − 285 , 206.5 , − 429 c 69 , − 144 , 104.5 , − 217.7 , 106.5 , − 221 l 0 − 0 c 5.3 , − 9.3 , 12 , − 14 , 20 , − 14 H 400000 v 40 H 845.2724 s − 225.272 , 467 , − 225.272 , 467 s − 235 , 486 , − 235 , 486 c − 2.7 , 4.7 , − 9 , 7 , − 19 , 7 c − 6 , 0 , − 10 , − 1 , − 12 , − 3 s − 194 , − 422 , − 194 , − 422 s − 65 , 47 , − 65 , 47 z M 83480 h 400000 v 40 h − 400000 z "/ >< / s vg >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.1328 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.93 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 0.313 e m ; " >< / s p an >< s p an c l a ss = " minn er " > ⋯ < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< p > T hi s i s anin f ini t eser i es , an d i t s v a l u e i s n o t kn o w n e x a c tl y . < / p >< h 2 >< s t ro n g > Q : Wha t i s t h es i g ni f i c an ceo f t h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n s ? < / s t ro n g >< / h 2 >< p > A : T h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n s i s a f u n d am e n t a lp ro b l e min g eo m e t ry t ha t ha s b ee n s t u d i e df orce n t u r i es . I t ha s man y a ppl i c a t i o n s inma t h e ma t i cs , p h ys i cs , an d e n g in eer in g , an d i s ak eyco n ce pt in t h es t u d yo f g eo m e t r i cs ha p es an d t h e i r p ro p er t i es . < / p >< h 2 >< s t ro n g > Q : Ho w c an I u se t h es u m o f t h e a re a so f l a r g es t in scr ib e d n − g o n s in re a l − w or l d a ppl i c a t i o n s ? < / s t ro n g >< / h 2 >< p > A : T h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n sc anb e u se d inman yre a l − w or l d a ppl i c a t i o n s , s u c ha s :< / p >< u l >< l i >< s t ro n g > C o m p u t er − ai d e dd es i g n ( C A D ) < / s t ro n g >: T h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n sc anb e u se d t ocre a t eco m pl e xg eo m e t r i cs ha p es an d m o d e l s . < / l i >< l i >< s t ro n g > E n g in eer in g < / s t ro n g >: T h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n sc anb e u se d t o d es i g nan d o pt imi ze g eo m e t r i cs ha p es an d s t r u c t u res . < / l i >< l i >< s t ro n g > P h ys i cs < / s t ro n g >: T h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n sc anb e u se d t os t u d y t h e p ro p er t i eso f g eo m e t r i cs ha p es an d t h e i r b e ha v i or u n d er d i ff ere n t co n d i t i o n s . < / l i >< / u l >< h 2 >< s t ro n g > Q : Wha t a reso m eco mm o nmi s t ak es t o a v o i d w h e n w or kin g w i t h t h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n s ? < / s t ro n g >< / h 2 >< p > A : S o m eco mm o nmi s t ak es t o a v o i d w h e n w or kin g w i t h t h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n s in c l u d e :< / p >< u l >< l i >< s t ro n g > I n correc tl yc a l c u l a t in g t h es i d e l e n g t h o f t h e l a r g es tp oss ib l ere gu l a r n − g o n < / s t ro n g >: M ak es u re t o u se t h ecorrec t f or m u l a t oc a l c u l a t e t h es i d e l e n g t h o f t h e l a r g es tp oss ib l ere gu l a r n − g o n . < / l i >< l i >< s t ro n g > I n correc tl yc a l c u l a t in g t h e a re a o f t h e l a r g es tp oss ib l ere gu l a r n − g o n < / s t ro n g >: M ak es u re t o u se t h ecorrec t f or m u l a t oc a l c u l a t e t h e a re a o f t h e l a r g es tp oss ib l ere gu l a r n − g o n . < / l i >< l i >< s t ro n g > N o t co n s i d er in g t h e in f ini t eser i es < / s t ro n g >: T h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n s i s anin f ini t eser i es , an d i t s v a l u e i s n o t kn o w n e x a c tl y . M ak es u re t oco n s i d er t h e in f ini t eser i es w h e n w or kin g w i t h t h es u m o f t h e a re a so f t h e l a r g es t in scr ib e d n − g o n s . < / l i >< / u l >